Totally twisted Khovanov homology

Abstract

The author learned the Heegaard-Floer idea from John Baldwin while at the Mathematical Sciences Research Institute for the program on Homology theories of knots and links in the spring of 2010. While at MSRI, he stumbled on to the constructions in this paper while trying to understand what he was being told, completing the proof of invariance in fall of 2010. John Baldwin and Adam Levine have used this idea, in conjunction with a construction of C. Manolescu, to describe Ozsváth and Szabó's knot Floer homology using spanning trees of a link diagram, [1]. The author would like to thank John Baldwin for those conversations, as well as P. Ozsváth and Z. Szabó for the great idea. He would also like to thank Liam Watson, Matt Hedden, and Tom Mark for listening as he

Khovanov homology and binary dihedral representations for marked links

“…This spectral sequence is the framed instanton theory analogue of the spectral sequence in [10]. They moreover conjecture a relation between Hd(D, ω) and a twisted Khovanov homology similar to those in [2], [6], and [14], which is an invariant of links with marking data, and which also has a spectral sequence relating it to the framed instanton homology.…”

Two-fold quasi-alternating links, Khovanov homology and instanton homology

“…We work with F = Z 2 coefficients throughout. This invariant is a modification of Roberts' totally twisted Khovanov homology [Rob15] in the spirit of Jaeger [Jae13]. In particular, we use the viewpoint of dots on arcs as in [Jae13] instead of regions, which was used in [Rob15].…”

“…In the next section, we will introduce the δ-graded twisted Khovanov homology Kh(L, ω) of a two-fold marked link (L, ω) with Z 2 = Z 2 coefficients. This is a variant of Roberts' Totally Twisted Khovanov homology [Rob15] and the more general construction that appears in Jaeger [Jae13]. It is also related to Baldwin, Levine and Sarkar's Khovanov homology for pointed links [BLS].…”

Twisted skein homology

“…As in [11], [6], we label each arc f ∈ arc(T ) with a formal variable x f and form the polynomial ring…”